This ME3 wiki page has a list of possible race/class combinations, but by default only the human options are available. How are the other races unlocked?
This might require updating after the game releases, but in the demo, you can get additional race/class unlocks by purchasing "item packs" with credits you earn by playing the game.
In the demo version, there's a "Recruit Pack" that costs 5,000 cr, and a "Veteran Pack" that costs 20,000 cr. Think of these like a pack of cards you might buy for a collectible card game - you'll get a random assortment of several different things, including the possibility of unlocking a new character.